Existence and Stability Analysis of Spring-Block Model Solutions with Rate and State Friction

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1 Applied Mathematical Sciences, Vol. 7, 2013, no. 36, HIKARI Ltd, Existence and Stability Analysis of Spring-Block Model Solutions with Rate and State Friction Kodwo Annan Department of Mathematics & Computer Science Minot State University Minot, North Dakota 58707, USA Copyright c 2013 Kodwo Annan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A four-dimension parameter model associated with spring-block system with rate- and state- friction laws was established. Existence, stability and bifurcation of the model system were proved using spectral analysis. Stable solution existed at the origin while bifurcation occurred at any point of the 4-dimension parameter boundary. Across the threshold of instability there existed a range of unstable modes parameterized by the wave-number. Keywords: Spring-Block, Stability, Rate-State-Friction, Spectral, Existence 1 Introduction It is believed that there exist complex physical properties and behaviors in the earth s crust and along fault surfaces that prevent our understanding and accurately predicting earthquake dynamics [1, 2]. From this viewpoint, the Earth crust consists of tectonic plates which slip along existing faults relative to each other and subjected to friction laws of faults, also known as stickslip process [3]. The underlying physical behaviors and its phenomenology of velocity-weakening friction laws have been extensively studied using the spring-block models [4-7]. However, a wealth of laboratory-scale experimental

2 1786 Kodwo Annan friction data exists that could be used with spring-block models to describing fully these stick-slip processes. The concept of using laboratory determined friction laws (also known as the rate- and state- dependent friction laws) was first introduced by Dieterich and Ruina in the late 70s [8,9]. Dieterich devised an empirical law that described the behavior of friction coefficient (for both steady and transient states) based on rock friction experiment. Later, the formulation was modified by Ruina by introducing additional variables other than the sliding velocity (called the state variables). Thus, the laboratory measurements suggested that both the velocity dependence of the friction and the time dependent of the static friction were important in understanding and predicting earthquake dynamics. The basic idea underlying this spring-block models with rate- and statefriction (RSF) laws is that the system considers the effects of inertia and deformations through the introduction of masses and springs. The rate implies the instantaneous rate of deformation is dependent on the friction laws while the state suggests that the system s internal state depends on the friction laws. The aforementioned details of the system include such effects as, e.g., dependence of friction on environmental conditions (temperature, humidity, presence of lubricant), history of friction contact (memory effect), wear of materials and surface roughness. The RSF laws have been applied to fault models to investigate seismic cycles including pre-seismic slip and nucleation, growth of dynamic instabilities, earthquake after slip and after-shocks [10-13]. RSF laws have also been used in seismological studies to describe variations of seismicity rates and earthquake patterns [14-16]. Yet, the existence of periodic solutions to the non-linear rate and state spring-block model for finitely large enough time scales remain unexplained. In this study, we use spring-block model with fairly homogeneous RSF laws to analytically prove the existence and stability of periodic solutions when the model reclines at the threshold of instability. The rest of the paper is arranged as follows: The 4-dimension parameter that influence spring-block model with rate and state dependent friction laws was established in section 2. Spectral method was employed to establish the existence, stability and bifurcation of the model solutions in section 3. We then give the conclusion in section 4. 2 Model Formulation We adapt the dimensionless spring-block model derived in [6] for the Burridge and Knopoff model [17] where the friction force is replaced by rate- and state-

3 Existence and stability analysis of spring-block model solutions 1787 dependent friction laws and x i (t), θ i (t) R as ẍ i + x i l 2 (x i+1 2x i + x i 1 )=F (ẋ i,θ i ); i Z. (1) Since the state variable, θ i, is time-dependent, we must have a time evolution law for θ i together with the friction force. Many empirical laws have been proposed in order to describe the time-dependent properties of friction force. Here, we assume the dimensionless form of the Dieterich s law (also known as slowness law) [9] by θ i =1 θ i (V +ẋ i ). (2) Equation (2) describes the time-dependent increase of the state variable even at V = 0. Since the loading rate associated with the plate motion is typically a few centimeters per year, the dimensionless loading rate V is estimated as 10 8 for a characteristic slip distance greater than zero. Finally, we adapt and modify the dimensionless form of RSF force as given by [18] as F (ẋ i,θ i )=c + a log(1 + V +ẋ i )+b log θ i, (3) where a and b are positive dimensionless constants describing the RSF law and cis a reference friction coefficient at a reference sliding velocity. The dimensionless parameter c is estimated to fall between 10 3 and 10 4 while the parameters a and b are one or two orders of magnitude smaller than c. Introducing x i = x + X i, and θ i = θ + ψ i for i Z, where θ = V 1 and x = (c + a log(1 + V )+b log θ, equations (1)-(3) reduces to Ẍ i + X i = l 2 (X i+1 2X i + X i 1 ) aẋ i 1+V + aẋ2 i 2(1+V ) 2 aẋ3 i 3(1+V ) 3 bv ψ i + bv 2 ψi 2 bv 3 ψi 3 + o(ẋi,ψ 2 3 i ); ψ i = V 1 Ẋ i Vψ i ψ i Ẋ i ; for o(ẋi,ψ i )=o(x 4 )+o(ψ 4 ) Expanding the friction using Taylor expansion reduces equation (4) to. (4) { Ẍi + X i = l 2 (X i+1 2X i + X i 1 )+a 1 Ẋ i + b 1 ψ i +Γ(Ẋi,ψ i ); ψ i = α 1 Ẋ i + α 2 ψ i + α 3 ψ i Ẋ i, for i Z, (5) where Γ(Ẋi,ψ i )=a 2 X 2 +a 3 X 3 +o(x 4 )+b 2 ψ 2 +b 3 ψ 3 +o(ψ 4 ), a 1 = a(1+v ) 1, b 1 = bv and a j,b j,α j for j =1, 2, 3 are parameters. Since a 1,b 1,α 1 and α 2 are negative, we make the following Hypothesis a 1,b 1,α 1 and α 2 are non positive. (6) Thus, the system (5) depends on a four-dimension parameter P = (a 1,b 1,α 1,α 2 ).

4 1788 Kodwo Annan 3 Spectral Analysis We now perform spectral analysis on (5) to investigate the model solution s existence and stability. 3.1 Notation and Preliminaries If we linearize and rewrite equation (5) as a system of first order differential equations in Hilbert space ξ = l 2 (Z) l 2 (Z) l 2 (Z) by denoting ΔX = X i+1 2X i + X i 1, we have dv dt = LV +N(V ), for V = X Ẋ ψ ξ, (7) where X =(X i ) i Z, Ẋ =(Ẋi) i Z and ψ =(ψ i ) i Z are all elements of l 2 (Z). The linear operator L acting on ξ is given by L = (1 + l 2 Δ) a 1 b 1, (8) 0 α 1 α 2 while the non-linear part of (7), that is N, is given by 0 N( x) = N 1 N 2, for x 1 x = x 2 x 3 ξ, (9) where N 1 and N 2 are given by { N1 ( x) =n 1 (x 2 )+n 2 (x 3 ), N 2 ( x) =α 3 x 2 x 3, (10) and { n1 (x 2 )=a 2 x a 3 x o(x 4 2), n 2 (x 3 )=b 2 x b 3 x o(x 4 3). (11) Thus, for x ξ, we have both N 1 ( x), N 2 ( x) l 2 (Z). It is now necessary to define the resolvent and spectrum sets of a linear operator L on an infinitedimensional Hilbert space. Definition 3.1. The resolvent set of L ξ, denoted by ρ(l), is the set of complex numbers λ such that (L λi) :ξ ξ is one-to-one and onto.

5 Existence and stability analysis of spring-block model solutions 1789 Definition 3.2. The spectrum of L, denoted by σ(l), is the complement of the resolvent set in C, that is, σ(l) = C\ρ(L). If L λi is one-to-one and onto, then the open mapping theorem implies that (L λi) 1 is bounded. Hence, when λ ρ(l), both L λi and (L λi) 1 are one-to-one, onto and bounded linear operator. Thus, the operator L is a bounded operator from ξ into ξ. Proposition 3.3. The spectrum of a bounded operator L on a Hilbert space is nonempty. Proof. suppose that L ξ. Then the resolvent R λ =(λi L) 1 is an analytic function on ρ(l). Thus, for every x, y ξ, the function f : ρ(l) C defined by f(ρ) = x, R λy is analytic in ρ(l), and lim f(λ) = 0. Suppose, for λ contradiction, that σ(l) is empty. Then f is a bounded entire function, and Liouville s Theorem implies that f : C C is a constant function, so f =0. But if f =0 x, y ξ, then R λ =0 λ C, which is impossible. Hence σ(l) is not empty. 3.2 Existence of Periodic Solutions We first show with the following proposition that the spectrum σ(l) is continuous and represents the roots of polynomial (12) which is dependent on Hypothesis (6). Proposition 3.4. The spectrum of L is the set σ(l) ={λ C/ there exists r [0,π], P(λ, r) =0}, where the polynomial P (x, r) is of degree 3 in x depending on the wave number r [0,π], defined by { P (x, r) =x 3 (a 1 + α 2 )x 2 +(a 1 α 2 α 1 b 1 + δ r )x α 2 δ r, for δ r =1+4l 2 sin 2 (12) (0.5r). Proof. Since L is a linear operator in an infinite-dimensional Banach space, the spectrum of L are values of λ C such that L λi is non-injective or non surjective on ξ. So, if we let F =(L λ) x =(f 1,f 2,f 3 ) ξ for unknown x =(x 1,x 2,x 3 ), we have x 2 λx 1 = f 1, F =(L λ) x (l 2 Δ 1)x 1 +(a 1 λ)x 2 + b 1 x 3 = f 2, α 1 x 2 +(α 2 λ)x 3 = f 3, x 2 = f 1 + λx 1, x 3 = f 3 α 1 (f 1 +λx 1 ) [ α 2, λ ] λ(a 1 λ)λ + l 2 Δ+ b 1α 1 1 α 2 x λ 1 = [ ] λ a 1 + b 1α 1 α 2 f λ 1 + f 2 b 1 f α 2 λ 3. (13)

6 1790 Kodwo Annan To solve the third equation, we first make a Fourier Transform by denoting F : L 2 (T ) l 2 (Z) and f (C n (f)) n Z, where (C n (f)) n Z depicts the sequence of Fourier coefficients of a 2π periodic function f L 2. Thus, applying Plancherel s Theorem we have f L 2 (T ) = F(f) l 2 (Z). If we define F 1 : l 2 (Z) L 2 (T )to be the inverse bijection of F, then for a sequence a l 2 (Z), F 1 [a] is a unique function in L 2 (T ) and that a l 2 (Z) = F 1 [a] L 2 (T ). We then define an inverse bijection F 1 : Z ξ from F : ξ Z and state the following Lemma where F[ x] =(F[x 1 ] F[x 2 ] F[x 3 ]) 1. Lemma 3.5. For a l 2 (Z), [F 1 [Δa] = L(a) L 2 (T ), where L(a)(r) = 4 sin 2 (0.5r)F 1 [a]. Proof. Let g(a) 4 sin 2 (0.5r)ã where ã = F 1 [a] and L(a) =F 1 [Δa], then showing that g(a) = L(a) is the same as showing that F[ g(a)] = F[ L(a)] = Δa. Thus, F[ g(a)] = C n ( g(a)) = 1 2π 2π 4 sin 2 (0.5r)e inr ã(r)dr = 1 2(1 cos r)e inr ã(r)dr 0 2π 0 = 1 2π 2π 0 (eir + e ir )e inr ã(r)dr + 1 2π 2e inr ã(r)dr, 2π 0 = C n+1 (ã) 2C n (ã)+c n 1 (ã) =a n+1 2a n + a n 1 =Δa. 2π where Applying F 1 to (13) for λ = α 2, we have for F x 2 = f 1 + λ x 1, x 3 = f 3 α 1 ( f 1 +λ x 1 ) α 2, (14) λ P (λ, r) x 1 =(α 2 λ) f 2 +[b 1 α 1 (α 2 λ)(a 1 λ)] f 1 b 1 f3 { P (λ, r) =λ 3 (a 1 + α 2 )λ 2 +(a 1 α 2 α 1 b 1 + δ r )λ α 2 δ r, for δ r =1+4l 2 sin 2 (0.5r). The polynomial P (λ, r) depends on r only through sin 2 (0.5r), therefore, r is restricted to [0,π]. Also, if (14) has a solution in Z = [L 2 (T )] 3, then the solution must be unique. So, we deduce that there is no eigenvalues in the spectrum and that the spectrum is the set of λ C for which the operator L λi is non surjective. Therefore, we have the spectrum σ(l) = σ(r) and the resolvent set r [0,π] (T ). of L = λ x 1, x 2 and x 3 are in L 2 r (T ) for f 1, f 2, f 3 L 2 r For λ/ σ(l) {α 2 } and for all P (λ, r) 0, r [0,π], x 1 in equation (14) is given by x 1 (r) = [b 1α 1 (α 2 λ)(a 1 λ)] f 1 +(α 2 λ) f 2 b 1 f3. (15) P (λ, r)

7 Existence and stability analysis of spring-block model solutions 1791 We claim that if P (λ, r) is continuous and not equal to zero on the compact [0,π], then [P (λ, r)] 1 is bounded. That is there exists a constant M>0such that [P (λ, r)] 1 M for all r [0,r]. In other words ( ) f1 x 1 L 2 r C + f 2 + f 3 L 2 r L 2 r L 2 r P (λ,r) CM ( f1 L 2 r + P (λ,r) f 2 L 2 r + P (λ,r) f 3 L 2 r ) < Thus, x 1 L 2 r. Similarly from (14), x 2 and x 3 are also in L 2 r. Hence, λ is in the resolvent set of L. For λ = α 2, we have (L λ) x = F ξ α 2 x 1 = α 1 1 f 3 f 1 l 2 (Z) α 1 x 2 = f 3 l 2 (Z) b 1 x 3 = f 2 +(1+Δ)x 1 (a 1 α 2 )x 2 l 2 (Z) since α 1,α 2,b 1 0. Hence α 2 ρ(l). Finally, for λ σ(l), there exists a unique r 0 [0,π] such that P (λ, r 0 )= 0. Hence, x 1 has a singularity at r = r 0. From (15) we rewrite the polynomial P (λ, r) as P (λ, r) =h 1 (λ) sin 2 (0.5r)+h 2 (λ), where h 1 (λ) =4l 2 (λ α 2 ) and h 2 (λ) =P (λ, 0) = λ 3 (a 1 + α 2 )λ 2 +(1+ a 1 α 2 b 1 α 1 )λ α 2. Clearly, h 1 (λ) 0 since λ α 2. However, if r 0 0, then in the neighborhood of r 0 we have h 1 (λ) sin(0.5r 0 ) cos(0.5r 0 )(r r 0 ) P (λ, r 0 ). Implying, x 1 / L 2 r (T ). If r 0 =0, then we have P (λ, 0) = h 2 (λ) = 0 and P (λ, r) =h 1 (λ) sin 2 (0.5r) 0.25h 1 (λ)r 2. Thus x 1 / L 2 r (T ). Therefore, for λ σ(l), L λi is not surjective. Hence σ(l) is the roots of the polynomial P for which r [0,π]. 3.3 Stability and Bifurcation Analyses We now prove the existence of a bifurcation in our model system at some critical value of the hypothesis parameters (6) by Considering a possible bifurcation at some critical value of the parameter H = (a 1,b 1,α 1,α 2 ) R 4. Proposition 3.6. If we define the roots of the polynomial P (λ, r) for a fixed r [0,π] by λ R (r), λ (r) and λ + (r)and denote their regions and boundary in R 4 as: R = {H R 4 (a 1 α 2 α 1 b 1 )(a 1 + α 2 )+a 1 < 0} for region R, R + = {H R 4 (a 1 α 2 α 1 b 1 )(a 1 + α 2 )+a 1 > 0} for region R +, R c = {H R 4 (a 1 α 2 α 1 b 1 )(a 1 + α 2 )+a 1 =0} for boundary.,

8 1792 Kodwo Annan Then, for every wave number r [0,π] and the 4-dimension parameter H= (a 1,b 1,α 1,α 2 ) R 4, the polynomial function P (x, r) has one real non positive root λ R (r). In addition, if i H belongs to R, then for every r [0,π], λ ± (r)have non-positive real parts. ii H belongs to R +, then there exists a finite modes for which P (x, r) has positive real part roots iii H belongs to the boundary of R and R c, then for every r [0,π], λ ± (r)have non-positive real parts except at r =0when λ ± (r) are purely imaginary. Proof. For fixed r [0,π],P(x, r) is a polynomial of degree three and therefore should have at least a real roots λ R (r). In addition, from Hypothesis (6) P (α 2,r)= b 1 α 1 α 2 > 0 implying λ R (r) <α 2 < 0. We now show that the sign of the real parts of the roots of P (x, r) is non positive (that is Hurwitz polynomial) by considering the Routh-Hurwitz criterion. The Routh-Hurwitz criterion provides both necessary and sufficient condition to the Hurwitz polynomial. The criterion could be stated for our third degree polynomial P (x) =c 3 x 3 + c 2 x 2 + c 1 x + c 0 for c 3 > 0says P (x) is Hurwitz { ci > 0, for i =1, 2, 3 (RHC 1) c 1 c 2 c 0 c 3 > 0 (RHC 2) First, we observe that P (x, r) satisfies both conditions for all r. Ifδ>0: Condition RHC-1: Since c 3 =1> 0,c 2 = (a 1 + α 2 ) > 0 and c 0 = α 2 δ>0, we only need to show that c 1 > 0. It follows that a 1 α 2 α 1 b 1 + δ>0 r [0,π], a 1 α 2 α 1 b 1 +1> 0. (16) Condition RHC-2: c 1 c 2 c 0 c 3 > 0 (a 1 + α 2 )(a 1 α 2 α 1 b 1 + δ)+α 2 δ>0, δ>(a 1 + α 2 ) ( a 1 1 α ) 1b 1 α 2, a 1 α 2 α 1 b 1 + δ>a 1 1 α 2(α 1 b 1 a 1 α 2 ), a 1 α 2 α 1 b 1 +1> a 1 1 α (17) 2(a 1 α 2 α 1 b 1 ). We observe that if a 1 α 2 α 1 b 1 > 0, equation (16) and (17) are satisfied. When a 1 α 2 α 1 b 1 < 0, equation (17) implies (16). Therefore, P (x, r) has non positive real part roots for all r if and only if H satisfies (a 1 α 2 α 1 b 1 )(1+a 1 1 α 2 )+1 > 0. Thus, the origin is steadily stable if and only if H lies in R. Outside the region, we observe two things:

9 Existence and stability analysis of spring-block model solutions If H R c, then (16) is satisfied for all r and from (17), 4l 2 sin 2 (0.5r) > 0. Hence the mode r = 0 is critical since P (x, 0) is not a Hurwitz polynomial. 2. If H R +, then a 1 α 2 α 1 b 1 +1< a 1 1 α 2(a 1 α 2 α 1 b 1 ). Thus, there exists a band of modes r 1 [0,π] such that for every r [0,r 1 ] we have a 1 α 2 α 1 b l 2 sin 2 (0.5r) < a 1 1 α 2 (a 1 α 2 α 1 b 1 ). Therefore, for all r [0,r 1 ], the polynomial P (x, r) has positive real parts and thus, the modes are unstable. So, R c represents the threshold of instability for our system. Let assume now that we cross this critical threshold R c at a point Δ c = [a c 1,b c 1,α c 1,α c 2] and set a small parameter ε such that Δ = Δ(ε) =[a 1 (ε),b 1 (ε),α 1 (ε),α 1 (ε)] then Δ(ε) isinr + or R depending on the sign of ε 0 and Δ c R c at ε =0. So, at ε = 0 there is a change in stability at the origin. Theorem 3.7. At ε =0, the spectrum of L has a complex pair of element crossing the imaginary axis at ±iω with ω = a c 1 αc 2 αc 1 bc Proof. Recall that at ε =0, Δlies in the critical range of R c and that all the roots of P (λ, r)have non positive real part for r ]0,π]. Suppose r =0, then for P (λ, r) we have imaginary root λ = iω if and only if P (iω, 0) = 0 { ω 2 = a c 1 αc 2 αc 1 bc 1 +1 ω 2 = αc 2 α c 2 +ac 1 ω = ± a c 1 αc 2 αc 1 bc 1 +1, Since (a c 1 + α c 2)(a c 1 α c 1b c 1 α c 2) = 1 for Δ c R c. Thus specifically, at ε =0,P(x, r) has imaginary root for r = 0 and that all other roots have negative real parts. However, if we cross the boundary, there exists a group of wave-numbers r in which the roots of P (x, r) have positive roots. This implies that the group of modes, r, becomes unstable starting from when r = 0. Therefore, a bifurcation occurs at any point on Δ c =[a c 1,b c 1,α 1,α c 2]. c 4 Conclusion The parameters affecting the existence of periodic solutions generated with rate and state friction in spring-block model was studied. Spectral analysis was used to investigate the sign of the real parts of the roots of the system to establish the model s existence, stability and bifurcation. It was shown that

10 1794 Kodwo Annan there exist periodic traveling wave solutions close to the origin and a bifurcation at some critical value of the dependent parameters. The system s origin was uniformly stable while across the threshold of instability there existed a wide range of unstable modes. References 1. C. H. Scholz, The mechanics of earthquakes and faulting, Cambridge University Press, Cambridge, C. H. Scholz, Earthquakes and friction laws, Nature, 391, No (1998) B. F. Feeny, A. Guran, N. Hinrichs, and K. Popp, A historical review of dry friction and stick-slip phenomena, Applied Mechanics Reviews 51 (5) (1998). 4. J. M. Carlson, J. S. Langer, and B. E. Shaw, Dynamics of earthquake faults, Reviews of Modern Physics, 66, 657, U. Galvanetto, Sliding bifurcations in the dynamics of mechanical systems with dry friction remarks for engineers and applied scientists, J. Sound Vibr., 276, (2004). 6. K. Annan, Propagation of local and global smoothed periodic waves in a spring-block model, IJPAM (2012) Vol. 81 No. 3, J. R. Rice, and A. L. Ruina, Stability of Steady Frictional Slipping, J. Appl. Mech.50, (1983). 8. J. H. Dieterich, Modeling of rock friction: Experimental results and constitutive equations, J. Geophys. (1979), Res., 84(B5), , doi: /jb084ib05p A. Ruina, Slip instability and state variable friction laws, J. Geophys. (1983), Res., 88(B12), 10,359 10,370, doi: /jb088ib12p S. T. Tse, and J. R. Rice (1986), Crustal earthquake instability in relation to the depth variation of frictional slip properties, J. Geo phys. Res., 91, J. R. Rice, Spatiotemporal complexity of slip on a f ault,j. Geophys. Res., (1993), 98, C. Marone, Laboratory-derived friction laws and their application to seismic faulting, Annu. Rev. Earth Planet. (1998), Sci., 26,

11 Existence and stability analysis of spring-block model solutions J. H. Dieterich, J. H. (1994), A constitutive law for rate of earthquake production and its application to earthquake clustering, J. Geophys. Res., 99, J. H. Dieterich, V. Cayol, and P. Okubo (2000), The use of earthquake rate changes as a stress meter at Kilauea volcano, Nature, 408, S. Toda, R. S. Stein, and T. Sahiya (2002), Evidence from the AD 2000 Izu islands earthquake swarm that stressing rate governs seismicity, Nature, 419, R. S. Stein, (2003), Earthquake conversations, Sci. Am., 288, R. Burridge and L. Knopoff, Model and theoretical seismicity, Bull. Seism. Soc.Am. 57, (1967). Received: January, 2013